In The Crucible, why does Elizabeth Proctor fire Abigail?

Elizabeth fires Abigail because she knows that the girl
and her husband were having an affair.  It is the source of all the trouble that brews
in Salem.  In exchange for being fired, Abigail and the girls dance naked in the woods,
while she drinks a potion concocted by Tituba as an attempt to "remove" Elizabeth so
that she can be with John Proctor.

It is shared early on in
the drama that Abigail bears resentment for being "sent out" by Elizabeth.  She says so
the first moment she and John are alone in the second scene of the first
Act:

I know
how you clutched my back behind your house and sweated like a stallion whenever I come
near!  Or did I dream that?  It's she [Elizabeth] put me out, you cannot pretend it were
you.  I saw your face when she put me out, and you loved then and you do
now!

There is resentment that
Abigail feels for being fired by Elizabeth.  However, it is clear that there was an
affair between the both of them.  In this, Elizabeth acts in the name of her
marriage.

We get some level of insight in to Elizabeth's
motivation for firing Abigail from her own words.  In the first scene of Act II, it is
alluded that she knew of the relationship and the feelings that John harbored for the
girl and might still.  When she suggests that "the magistrate" in John's heart "sits in
judgment," we get the idea that the firing was needed in order to begin the process of
moral reconciliation, something that is still ongoing in the early phases of the
drama.

The courtroom scene of Act III brings us the most
direct words from Elizabeth as to why she had to fire Abigail.  She is summoned at the
point where Proctor has confessed to his adultery with Abigail, in the attempt to
discredit her and the accusations that have escalated to an out of control condition. 
Danforth "forces" her to disclose the reason why Abigail was
fired:

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Danforth:  Why did you dismiss Abigail
Williams?

Elizabeth:  She dissatisfied me.
(Pause) And my husband.

Danforth:  In
what way dissatisfied you?

Elizabeth:  She were...Your
honor, I- in that time I was sick.  And I- My husband is a good and righteous man.  He
is never drunk as some are, nor wastin' his time at the shovelboard, but always at  his
work.  But in my sickness- you see, sir, I were a long time sick after my last bby, and
I thought I saw my husband somewhat turning from me.  And this girl- (She
turns to
Abigail.)

This is
the closes that Elizabeth comes from admitting that the reason for the dismissal was
because of the affair.  It is at this critical moment, her "crucible" as it were, does
Elizabeth turn from the truth and lie in court that her husband did not have an affair,
something that he already confessed in public.  Demonstrating how a corrupt legal system
can hardly be trusted in finding the truth, Elizabeth's real reason for dismissing
Abigail was for the affair with her husband remains in her heart and is not spoken into
the legal record.